ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS. UNIT 2 - Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS. 2 hours 30 minutes

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1 TEST CODE FORM TP MAY/JUNE 2012 CARIBBEAN EXAMINATIONS COUNCIL ADVANCED PROFICIENCY EXAMINATION PURE MATHEMATICS UNIT 2 - Paper 02 ANALYSIS, MATRICES AND COMPLEX NUMBERS 2 hours 30 minutes ( 25 MAY 2012 (p.m.) ) This examination paper consists of THREE sections: Module 1, Module 2 and Module 3. Each section consists of 2 questions. The maximum mark for each Module is 50. The maximum mark for this examination is 150. This examination consists of 7 printed pages. READ THE FOLLOWING INSTRUCTIONS CAREFULLY. 1. DO NOT open this examination paper until instructed to do so. 2. Answer ALL questions from the THREE sections. 3. Write your solutions, with full working, in the answer booklet provided. 4. Unless otherwise stated in the question, any numerical answer that is not exact MUST be written correct to three significant figures. Examination Materials Permitted Graph paper (provided) Mathematical formulae and tables (provided) - Revised 2012 Mathematical instruments Silent, non-programmable, electronic calculator īiiiiiiiii --==īiiiiiiiii == - Copyright 2011 Caribbean Examinations Council All rights reserved. DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.

2 ~ ~ SECTION A (Module 1) Answer BOTH questions. 1. (a) (i) Given the curve y = x' e', a) b) find the x-coordinates of the points at which ~ = 0 c) find the x-coordinates of the points at which ~ = 0 (ii) Hence, determine if the coordinates identified in (i) b) and c) above are at the maxima, minima or points of inflection of y = r- e'. [7 marks] (b) A curve is defined by the parametric equations x = sin:' -rt, y = f - 2t. Find (i) the gradient of a tangent to the curve at the point with parameter t (ii) the equation ofthe tangent at the point where t = ~. [3 marks]

3 (a) (i) Express X2_ 3x --'--::"::":"'-- (x-1)(x 2 + 1) in partial fractions. [7 marks] (ii) Hence, find (b) (i) Given that sin A cos B - cos A sin B = sin (A - B) show that cos 3x sin x = sin 3x cos x - sin 2x. (ii) If I = fcos m x sin 3x dx and m J = fcos m x sin 2x dx, m prove that (m + 3) 1 m = mj m _ 1 - cos" x cos 3x. [7 marks] (iii) Hence, by putting m = 1, prove that n 4 f"4 cos x sin 3x dx = f"4 sin 2x dx o 0 n (iv) Evaluate f"4 sin 2x dx. o n

4 -4- SECTION B (Module 2) Answer BOTH questions. 3. (a) For a particular G.P., u 6 = 486 and u ll = , where un is the nth term. (i) Calculate the first term, a, and the common ratio, r. (ii) Hence, calculate n if S = (b) The first four terms ofa sequence are 1 x 3, 2 x 4, 3 x 5,4 x 6. (i) Express, in terms ofr, the rth term, u,., of the sequence. [4 marks] I, J (ii) Prove, by mathematical induction, that U = _1 n (n+ 1) (2n + 7), vn E N. r 6 r= I [7 marks] (c) (i) Use Maclaurin's Theorem to find the first three non-zero terms in the power series expansion of cos 2x. (ii) Hence, or otherwise, obtain the first two non-zero terms in the power series expansion of sin" x. )

5 = (a) (i) Express [ ~ J in terms of factorials. [1 mark] (ii) Hence, show that [ ~ J ~ [n:.] [3 marks] (iii) Find the coefficient of x' in ~ - ;J [5 marks] (iv) Using the identity (1 + x)2n = (1 + x)" (1 + x)", show that [2 n nj ~ c; + c; + c; c; - I + c;, where < ~ [ ~J. [8 marks] (b) Letf(x) = 2x 3 + 3x 2-4x - 1 = O. (i) Use the intermediate value theorem to determine whether the equationf(x) has any roots in the interval [0.2, 2]. (ii) Using x. = 0.6 as a first approximation ofaroot T off(x), execute FOUR iterations of the Newton-Raphson method to obtain a second approximation, x 2, oft.

6 -6- SECTION C (Module 3) Answer BOTH questions. 5. (a) How many 4-digit even numbers can be formed from the digits 1,2,3,4,6, 7, 8 (i) if each digit appears at most once? [4 marks] (ii) ifthere is no restriction on the number of times a digit may appear? [3 marks] (b) A committee of five is to be formed from among six Jamaicans, two Tobagonians and three Guyanese. (i) Find the probability that the committee consists entirely of Jamaicans. [3 marks] (ii) Find the number of ways in which the committee can be formed, given the following restriction: There are as many Tobagonians on the committee as there are Guyanese. (c) Let A be the matrix [ 1 0 3J (i) Find the matrix B, where B = A2-3A - I. [3 marks] (ii) Show that AB = -91. [1 mark] (iii) Hence, find the inverse, A~I, of A. (iv) Solve the system oflinear equations [3 marks]

7 -7-6. (a) (i) Draw the points A and B on an Argand diagram, 1+i {2 where A = -1-' and B = -1-' (ii) Hence, or otherwise, show that the argument of (1 +1~ + i) is EXACTLY 3n. -/ 8 (b) (i) Find ALL complex numbers, z, such that z' = i. [3 marks] (ii) Hence, find ALL complex roots of the equation Z2 - (3 + 5i) Z - (4-7i) = O. (c) Use de Moivre's theorem to show that cos 6 ()= cos'' () - 15 cos" () sirr' () + 15 cos' ()sin" () - sin" (). END OF TEST IF YOU FINISH BEFORE TIME IS CALLED, CHECK YOUR WORK ON THIS TEST.